epsilon

arXiv:hep-ph/9906272v1 7 Jun 1999

DISSIPATIVE CONTRIBUTIONS TO ε ′/ε

F. Benatti Dipartimento di Fisica Teorica, Universit`a di Trieste Strada Costiera 11, 34014 Trieste, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste

R. Floreanini Istituto Nazionale di Fisica Nucleare, Sezione di Trieste Dipartimento di Fisica Teorica, Universit`a di Trieste Strada Costiera 11, 34014 Trieste, Italy

Abstract The time evolution and decay of the neutral kaon system can be described using quantum dynamical semigroups. Non-standard terms appear in the expression of relevant observables; they can be parametrized in terms of six, new phenomenological constants. We discuss how the presence of these constants affects the determination of the parameter ε′ /ε.

1

The neutral kaon system can be viewed as a specific example of an open quantum system.[1-3] Its propagation and decay are described by time-evolutions that generalize the standard approach, keeping into account possible new effects due to dissipation and loss of coherence;[4-6] these extended evolutions form a so-called quantum dynamical semigroup. The physical motivation behind this description comes from the observation that the quantum fluctuations of the gravitational field at Planck’s scale should produce loss of quantum coherence.[7] The origin of these effects can be ultimately traced to the dynamics of strings; indeed, one can show that a generalized, dissipative dynamics for the neutral kaon system can be the result of its weak interaction with a gas of D0-branes, effectively described by a heat-bath of quanta obeying infinite statistics.[8] Nevertheless, the general structure of the extended neutral kaon dynamics based on quantum dynamical semigroups is essentially independent from the actual “microscopic” mechanism that drives the dissipative effects. Indeed, the form of the evolution equation is fixed by basic physical requirements, like forward in time composition (semigroup property), probability conservation, entropy increase and complete positivity. In particular, this last property is crucial for the consistency of the generalized time evolution, since it guarantees, also for correlated systems, the positivity of the probabilities (for a complete discussion, see [9]). A rough dimensional estimate reveals that the new, dissipative effects are expected to be very small, at most of order m2K /MP ∼ 10−19 GeV, where mK is the kaon mass, while MP is Planck’s mass. Nevertheless, a detailed study of the K 0 -K 0 system using quantum dynamical semigroups shows that the dissipative contributions to the extended dynamics should be in the reach of the next generation of dedicated kaon experiments.[5, 6] In view of these results, in the following we shall analyze in detail how the non-standard dissipative contributions could affect the determination of the parameter ε′ /ε. As usual, we shall model the time-evolution and decay of the K 0 -K 0 system by means of a two-dimensional Hilbert space.[10] Since we are discussing phenomena leading to possible loss of quantum coherence, kaon states will be described by a density matrix ρ. This is a positive hermitian operator, i.e. with positive eigenvalues, and constant trace (for unitary evolutions). In the basis |K1 i and |K2 i of definite CP , one thus has:   ρ1 ρ3 ρ= , ρ4 ≡ ρ∗3 . (1) ρ4 ρ2 The evolution in time of this matrix can be described in general by an equation of the following form: ∂ρ(t) † = −iHeff ρ(t) + iρ(t) Heff + L[ρ] . (2) ∂t The effective hamiltonian Heff (the Weisskopf-Wigner hamiltonian) includes a nonhermitian part, that characterizes the natural width of the states: Heff = M − 2i Γ . The entries of these matrices can be expressed in terms of the complex parameters ǫS , ǫL , appearing in the eigenstates of Heff ,     1 1 1 ǫL , |KL i = , (3) |KS i = 2 1/2 2 1/2 ǫS 1 (1 + |ǫS | ) (1 + |ǫL | ) 2

and the four real parameters, mS , γS and mL , γL , the masses and widths of the states in (3), characterizing the eigenvalues of Heff : λS = mS − 2i γS , λL = mL − 2i γL . It proves convenient to use also the following positive combinations: ∆Γ = γS − γL , ∆m = mL − mS , as well as the complex quantities: Γ± = Γ ± i∆m and ∆Γ± = ∆Γ ± 2i∆m, with Γ = (γS + γL )/2. The additional piece L[ρ] generates a trace-preserving, positive, dissipative map, i.e. with increasing entropy S[ρ(t)] = −Tr[ρ(t) log ρ(t)], whose form is uniquely fixed by the requirement of complete positivity. To write it down explicitly, one expands ρ in terms of Pauli matrices σ1 , σ2 , σ3 and the identity σ0 : ρ = ρµ σµ , µ= 0,1, 2, 3. In this way, the map L[ρ] can be represented by a symmetric 4 × 4 matrix Lµν , acting on the column vector with components (ρ0 , ρ1 , ρ2 , ρ3 ). It can be parametrized by the six real constants a, b, c, α, β and γ:[4]   0 0 0 0   0 a b c  (4) Lµν = −2   , 0 b α β 0 c β γ with a, α and γ non-negative. These parameters are not all independent; to assure the complete positivity of the time evolution generated by (2), they have to satisfy certain inequalities (for details, see [4, 6]). Physical properties of the neutral kaon system can be computed from the density matrix ρ(t), solution of (2), by taking its trace with suitable hermitian operators. As already mentioned, the parameters a, b, c, α, β and γ can be assumed to be small, of order 10−19 GeV, roughly of the same order of magnitude of ǫS ∆Γ and ǫL ∆Γ; therefore, one can use an expansion in all these small parameters and approximate expressions for the entries of ρ(t) can be obtained (see [4]). In particular, one can work out the contributions ρL and ρS that correspond to the physical “long lived” and “short lived” neutral kaon states:   ǫL + ρL = NL 

2 2C ∗ ∆Γ−

+

γ ∆Γ

C 2 − 8 ∆Γ+ − 4 Re

ǫ∗L

+

2C ∆Γ+

and 

ρS = NS 

ǫ∗S +

1 ǫS +

2C ∆Γ−

ǫS +

2 2C ∆Γ−

−

γ ∆Γ

ǫL C ∆Γ




ǫL +

2C ∗ ∆Γ−

1

2C ∗ ∆Γ+

C 2 − 8 ∆Γ − 4 Re +

ǫS C ∗ ∆Γ






  , 

 ,

(5)

(6)

where C = c + iβ and NL , NS are suitable normalization constants, whose explicit expressions are irrelevant for the discussion that follows. Note that the states ρL and ρS are mixed: the matrices (5) and (6) are not projectors; only in absence of the contribution in (4) one recovers the matrices |KL ihKL | and |KS ihKS |. Useful observables are associated with the decays of the neutral kaons, in particular into pion states. The amplitudes for the decay of a K 0 state into π + π − and 2π 0 final states are usually parametrized as follows, in terms of the s-wave phase-shifts δi and the 3

complex amplitudes Ai , Bi , i = 1, 2: [11] 1 A(K 0 → π + π − ) = (A0 + B0 ) eiδ0 + √ (A2 + B2 ) eiδ2 , 2 √ A(K 0 → 2π 0 ) = (A0 + B0 ) eiδ0 − 2 (A2 + B2 ) eiδ2 ,

(7a) (7b)

where the indices 0, 2 refers to the total isospin I. The amplitudes for the K 0 decays are obtained from these with the substitutions: Ai → A∗i and Bi → −Bi∗ . The imaginary parts of Ai signals direct CP -violation, while a non zero value for Bi will also break CP T invariance. To construct the operators that describe these two pion decays in the formalism of density matrices, one has to pass to the basis |K1 i, |K2 i of definite CP . It is convenient to label the corresponding decay amplitudes as follows: A(K1 → π + π − ) = X+− , A(K1 → 2π 0 ) = X00 ,

A(K2 → π + π − ) = Y+− A(K1 → π + π − ) ,

A(K2 → 2π 0 ) = Y00 A(K1 → 2π 0 ) .

(8)

The complex parameters X and Y can be easily expressed in terms of Ai , Bi and δi of (7). For the dominant amplitudes, one easily finds: i h i √ h X+− = 2 Re(A0 ) + iIm(B0 ) eiδ0 + Re(A2 ) + iIm(B2 ) eiδ2 , (9a) i h i √ h (9b) X00 = 2 Re(A0 ) + iIm(B0 ) eiδ0 − 2 Re(A2 ) + iIm(B2 ) eiδ2 . On the other hand, the K2 amplitudes are suppressed, since they involve only CP and CP T violating terms. For the considerations that follow, it will be sufficient to work in the approximation that keeps only the dominant terms in these violating parameters. Then, the amplitude ratios Y+− and Y00 can be written as: Y+− = ε − ǫL + ε′ ,

Y00 = ε − ǫL − 2ε′ ,

where the parameters ε and ε′ take the familiar expressions:     ǫL + ǫS Im(A0 ) ǫL − ǫS Re(B0 ) ε= + , +i + 2 Re(A0 ) 2 Re(A0 ) and

    iei(δ2 −δ0 ) Re(A2 ) Im(A2 ) Im(A0 ) Re(B0 ) Re(B2 ) √ ε = +i . − − Re(A0 ) Re(A2 ) Re(A0 ) Re(A0 ) Re(A0 ) 2 ′

(10)

(11)

(12)

The first, second square brackets in (11) and (12) contain the CP , respectively CP T , violating parameters arising from the neutral kaon mass and decay matrices, as well as from their decay amplitudes. The factor ω = Re(A2 )/Re(A0 ) corresponds to the suppression due to the ∆I = 1/2 rule in the kaon two-pion decays. This ratio is known to be small; therefore, in presenting the above formulas, first order terms in the small parameters 4

multiplied by ω 2 have been consistently neglected. Note that the expressions for ε and ε′ given in (11) and (12) are independent from the phase conventions adopted in describing the kaon states. Using (8), the operators that describe the π + π − and 2π 0 final states and include direct CP and CP T violations are readily found:     1 Y00 1 Y+− 2 2 . (13) , O00 = |X00 | O+− = |X+− | ∗ ∗ Y00 |Y00 |2 Y+− |Y+− |2 Similar results can be obtained for other decay modes; for explicit expressions, see [5, 6]. With the help of these matrices, one can now compute various useful observables of the neutral kaon system. For example, the decay rates of the the physical states KL and KS , represented by ρL and ρS , into π + π − are simply given by the traces of (13) with the density matrices (5), (6)   A(KL → π + π − ) 2 ≡ Tr O+− ρL = |X+− |2 NL RL , (14a) +−   A(KS → π + π − ) 2 ≡ Tr O+− ρS = |X+− |2 NS RS , (14b) +− where, up to second order in the small parameters, 2   C 2 γ 2C ∗ ǫL C L , + Y+− + − 8 R+− = ǫL + − 4 Re ∆Γ− ∆Γ ∆Γ+ ∆Γ    2C S R+− = 1 + 2 Re ǫS + Y+− . ∆Γ−

(15a) (15b)

Similar expressions hold for the decay into 2π 0 ; one just needs to make the substitution {+−} → {00} in the formulas above. The results in (14) and (15) differ from the standard ones due to the presence of the non-standard parameters c, β (via the complex combination C) and γ. As a consequence, the double ratio: , A(KL → 2π 0 ) 2 A(KL → π + π − ) 2 (16) A(KS → 2π 0 ) ≡ 1 + 6 Σ , A(KS → π + π − )

is no longer connected in a simple way to the parameter ε′ /ε. Indeed, to leading order in CP and CP T violation and ∆I = 1/2 enhancement, using (10), one finds:
 ′ 2 ε |ε| + 2 Re ε C/∆Γ+ Σ = Re ε |ε|2 + D
(17)  ′ Im ε C/∆Γ+ ε − 2 Im , ε |ε|2 + D where

    C 2 εC ǫL C γ + 4 Re − 4 Re . −4 D= ∆Γ ∆Γ+ ∆Γ+ ∆Γ 5

(18)

Only when c = β = γ = 0, the expression in (17) reduces to the standard result: Σ = Re(ε′ /ε). This observation is of relevance in view of the fact that until now experimental informations on ε′ /ε rely on the measure of the double ratio in (16). On general grounds, one expects the phase of ε′ to be very close to that of ε,[11] so that Im(ε′ /ε) should be small when compared with R(ε′ /ε); taking this as a working assumption, one can then neglect the contribution from Im(ε′ /ε) in (17). Even in this case, an experimental measure of the quantity Σ will result in a meaningful determination of R(ε′ /ε) provided estimates on the non-standard dissipative parameters c, β and γ are independently given. Fortunately, by exploiting the explicit time-dependence of various neutral kaon observables that are directly accessible to the experiment, one can get independent evaluations for all the dissipative parameters appearing in (17), (18). For example, the decay rate for a pure K 0 initial state is given by: h i Tr ρK 0 (t)O+− i R+− (t) = h (19) Tr ρK 0 (0)O+− L = e−γS t + R+− e−γL t + 2 e−Γt |η+− | cos(∆m t − φ+− ) ,

L where R+− is the already encountered two-pion decay rate for the KL state, while the combination

η+− ≡ |η+− |eiφ+− = ǫL + Y+− −

2C ∗ 2C ∗ = ε + ε′ − , ∆Γ− ∆Γ−

(20)

L determines the interference term. Both R+− and η+− have been measured using different experimental setups.[12, 13] Other interesting observables, are the asymmetries associated with the decay into the final state f of an initial K 0 as compared to the corresponding decay into f¯ of an initial K 0 . All these asymmetries have the general form i h i h 0 Tr ρK¯0 (t)Of¯ − ρK (t)Of i h i . A(t) = h (21) Tr ρK¯0 (t)Of¯ + ρK 0 (t)Of L As discussed in [5], combining the expressions of R+− and Re(η+− ) in (15a) and (20) with asymmetries involving both three-pion and semileptonic final states one is able to obtain bounds on some of the the dissipative parameters using available experimental data. Since the appearance of [5], new experimental results have become available, in particular from the CPLEAR Collaboration.[14-16] Updating the discussion in [5] with the new data and the revised world averages for the parameters ∆Γ and ∆m,[17] one finally obtains the following new estimates:

c = (0.7 ± 1.2) × 10−17 GeV ,

β = (−0.7 ± 1.3) × 10−17 GeV , γ = (0.1 ± 22.0) × 10−20 GeV . 6

(22)

Although compatible with zero, these figures can still be used to give a rough evaluation of the correction to the familiar relation Σ = Re(ε′ /ε), due to the dissipative parameters. It should be noticed that the results in (22) have been obtained neglecting direct CP violating terms, since they are too small to affect the above figures for c, β and γ; this approximation is perfectly adequate when computing the dissipative contributions in (17), since they are already multiplied by a factor ε′ . Furthermore, the validity of the ∆S = ∆Q rule has been also assumed, and direct CP T violating effects in the decay products have been ignored. With these approximations, ǫL ≃ ε can be deduced from the measure of the parameter η+− in (20). By writing the relation between the experimentally measured quantity Σ and Re(ε′ /ε) as Re (ε′ /ε) = Σ (1 − X ) , (23) using the above estimates for the the dissipative, non-standard parameters, one can obtain a preliminary rough bound on the correction term

X =

C 2 4 ∆Γ + 4 Re +

  εC − − 2 Re ∆Γ +   εC |ε|2 + 2 Re ∆Γ + ǫL C ∆Γ




γ ∆Γ

.

(24)

One explicitly gets: X = 0.4 ± 1.8 .

(25)

The accuracy of the present experimental results on the neutral kaon system is not high enough to allow a precise determination of the correction X , which is clearly compatible with zero. However, the next-generation of dedicated kaon experiments should provide more complete and precise data, so that the contributions of dissipative effects to the neutral kaon dynamics and in particular to the determination of ε′ /ε, if present, could be directly measured. If these new experiments will provide evidence for a nonvanishing X , the actual value of ε′ /ε could be significantly different from the measured value of the quantity Σ. The standard model predicts ε′ /ε to be non-zero; however, a precise theoretical determination of its value is difficult due to cancellations in some hadronic contributions. The theoretical estimates vary from a few times 10−4 to about 10−3 , although with large uncertainties.[18-20] On the contrary, the KTEV Collaboration has recently announced a new, very precise measurement of the double ratio in (16), giving: Σ = (2.80 ± 0.41) × 10−3 .[21] Although various mechanisms have been advocated to solve this apparent discrepancy,[22-24] it is intriguing to observe that the presence of dissipative effects in the kaon dynamics could account, via the combination (24), for the difference between the measured Σ and the theoretically predicted Re(ε′ /ε). Actually, one can turn the argument around and get an evaluation of the extra correction X using as inputs the theoretical evaluations of Re(ε′ /ε) and the above experimental value for Σ. Although from X alone one can not directly extract informations on the single parameters c, β and γ, a non-vanishing value of X would clearly signal the presence of dissipative effects in the neutral kaon dynamics. 7

The theoretical estimates on Re(ε′ /ε) are based on standard (perturbative and nonperturbative) quantum field techniques, and ultimately reduce to the evaluation of transition matrix elements of certain effective operators between kaon and two-pion states. In principle, the dissipative, non-standard phenomena, whose effects have been included in the dynamical evolution (2) via the additional term (4), can also contribute directly to the decay process of the neutral kaon system; as a consequence, without further analysis the theoretical prediction of Re(ε′ /ε) should not be used to obtain a reliable estimate of X , since additional dissipative contributions might have been overlooked. Fortunately, the effects of dissipative phenomena to the kaon decay processes have been analyzed, and found to be negligible for any practical considerations;[25] in other words, the kaon decay is driven by the weak interaction and not by dissipative “gravitational” or “stringy” effects. Therefore, at least in principle, bounds on  the existence of the correction X can be ′ obtained, by rewriting (23) as: X = 1 − Re(ε /ε) Σ. The most accurate theoretical determinations of Re(ε′ /ε) are based on lattice computations,[18] phenomenological approaches,[19] and the chiral quark model.[20] Using the KTEV measure of Σ, from the results for Re(ε′ /ε) given in Refs.[18, 19, 20], one respectively finds: ( +0.16 0.65 −0.29 +0.68 +0.15 , X = 0.22 −0.44 . (26) X = 0.79 −0.21 , X = +0.13 0.76 −0.22 The errors in these determinations are based on flat distributions, and give just a rough idea of the accuracy of the central values. Although the figures in (26) seem to point towards a non-vanishing X , one should not regard this result as an evidence of dissipative effects in the dynamics of the neutral kaons. Rather, it should be considered as a rough evaluation of the sensitivity of the present knowledge of the parameter ε′ /ε in testing quantum dynamical time evolutions of the form given in (2). To improve the bounds in (26), more accurate inputs are necessary. Systems of two correlated neutral kaons, coming from the decay of a φ-meson, are also particularly suitable for the study of direct CP violation phenomena, in view of the possibilities offered by φ-factories. The φ-meson has spin one, and therefore its decay into two spinless bosons produces an antisymmetric spatial state. In the φ rest frame, the two neutral kaons are produced flying apart with opposite momenta; in the basis |K1 i, |K2 i, the resulting state can be described by:  1  (27) |ψA i = √ |K1 , −pi ⊗ |K2 , pi − |K2 , −pi ⊗ |K1 , pi . 2

The corresponding density operator ρA is a now 4 × 4 matrix, and its time-evolution can be obtained by assuming that, once produced in a φ decay, the kaons evolve in time each according to the completely positive map generated by the equation (2).[6] The typical observables that can be studied in such physical situations are double decay rates, i.e. the probabilities that a kaon decays into a final state f1 at proper time τ1 , while the other kaon decays into the final state f2 at proper time τ2 : h  i G(f1 , τ1 ; f2 , τ2 ) ≡ Tr Of1 ⊗ Of2 ρA (τ1 , τ2 ) ; (28) 8

as before, the operators Of1 and Of2 are the 2 × 2 hermitian matrices describing the final decay states. However, much of the analysis at φ-factories is carried out using integrated distributions at fixed time interval τ = τ1 − τ2 . One then deals with single-time distributions, defined by: Γ (f1 , f2 ; τ ) =

Z

∞ 0

dt G(f1 , t + τ ; f2 , t) ,

τ >0.

(29)

Starting with these integrated probabilities, one can form asymmetries that are sensitive to various parameters in the theory. In particular, in the case of two-pion final states, the following observable is particularly useful for the determination of ε′ /ε:[26] Aε′ (τ ) =

Γ (π + π − , 2π 0 ; τ ) − Γ (2π 0 , π + π − ; τ ) . Γ (π + π − , 2π 0 ; τ ) + Γ (2π 0 , π + π − ; τ )

(30)

Indeed, one can show that, to first order in all small parameters:  ε′  N (τ )  ε′  N (τ ) 1 2 Aε′ (τ ) = 3 Re − 3 Im . ε D(τ ) ε D(τ )

(31)

The τ -dependent coefficients N1 , N2 and D are functions of ε and of the dissipative parameters c, β and γ; their explicit expressions are collected in the Appendix. The clear advantage of using the asymmetry Aε′ to determine the value of ε′ /ε in comparison to the double ratio (16) is that, in principle, both real and imaginary part can be extracted from the time behaviour of (31). Due to the presence of the dissipative parameters however, this appears to be much more problematic than in the standard case; once more, a meaningful determination of ε′ /ε is possible provided independent estimates on c, β and γ are obtained from the measure of other independent asymmetries. This is particularly evident if one looks at the large-time limit (τ ≫ 1/γS ) of (31); indeed, in this limit, the expression of Aε′ /3 coincides with that of Σ given in (17), and the considerations presented there in discussing the case of a single kaon system apply also here. In conclusion, dissipative effects in the dynamics of both single and correlated neutral kaon systems could affect the precise determination of the ratio ε′ /ε; future dedicated experiments on kaon physics will certainly provide stringent bounds on these dissipative effects, thus clarifying their role in the analysis of the direct CP violating K 0 -K 0 piondecays.

9

Appendix For completeness, we collect here the explicit form of the three coefficients N1 , N2 and D that appear in the expression (31) for the asymmetry Aε′ ; they have been discussed also in [6], although using different approximations.      ε C γL + Γ− εC 2 −γS τ |ε| − 2 Re −e |ε| + 2 Re ∆Γ+ ∆Γ+ γS + Γ+       εC 2Γ −i∆mτ ε C −Γτ − Im sin(∆mτ ) (A.1) Re e − 4e ∆Γ+ γS + Γ+ ∆Γ+

−γL τ

N1 (τ ) = e



2

    εC ε C γL + Γ− −γS τ N2 (τ ) = 2e Im + 2e Im ∆Γ+ ∆Γ+ γS + Γ+        εC 2Γ 2ε C 2 −i∆mτ −Γτ + 2 Re sin(∆mτ ) |ε| + Im e − 2e ∆Γ+ γS + Γ+ ∆Γ+ (A.2) −γL τ

     C 2 ǫ C ε C γ L 2 − 4 Re + 4 Re − 4 |ε| + D(τ ) = e ∆Γ ∆Γ+ ∆Γ ∆Γ+ 2      C ǫL C γL γ − 8 − 4 Re + e−γS τ |ε|2 − γS ∆Γ ∆Γ ∆Γ+     C 2 ε C γL + Γ− 8 Γ(Γ + γ ) S − 4 Re + 4 −3 ∆Γ+ γS + Γ+ ∆Γ+ (Γ + γS )2 + (∆m)2     C 2 γL + Γ− 4εC 2Γ 2 −i∆mτ −Γτ |ε| + − 4 Re e − 2e ∆Γ+ γS + Γ+ ∆Γ+ γS + Γ+    εC cos(∆mτ ) − 4 Re ∆Γ+ −γL τ

10

(A.3)

References 1. R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Lect. Notes Phys. 286, (Springer-Verlag, Berlin, 1987) 2. V. Gorini, A. Frigerio, M. Verri, A. Kossakowski and E.C.G. Surdarshan, Rep. Math. Phys. 13 (1978) 149 3. H. Spohn, Rev. Mod. Phys. 53 (1980) 569 4. F. Benatti and R. Floreanini, Nucl. Phys. B488 (1997) 335 5. F. Benatti and R. Floreanini, Phys. Lett. B401 (1997) 337 6. F. Benatti and R. Floreanini, Nucl. Phys. B511 (1998) 550 7. S. Hawking, Comm. Math. Phys. 87 (1983) 395; Phys. Rev. D 37 (1988) 904; Phys. Rev. D 53 (1996) 3099; S. Hawking and C. Hunter, Phys. Rev. D 59 (1999) 044025 8. F. Benatti and R. Floreanini, Ann. of Phys. 273 (1999) 58, hep-th/9811196 9. F. Benatti and R. Floreanini, Mod. Phys. Lett. A12 (1997) 1465; Banach Center Publications, 43 (1998) 71; Comment on “Searching for evolutions of pure states into mixed states in the two-state system K-K”, hep-ph/9806450 10. T.D. Lee and C.S. Wu, Ann. Rev. Nucl. Sci. 16 (1966) 511 11. L. Maiani, CP and CP T violation in neutral kaon decay, in The Second Da Φne Physics Handbook, L. Maiani, G. Pancheri and N. Paver, eds., (INFN, Frascati, 1995) 12. C. Geweniger et al., Phys. Lett. B48 (1974) 487 13. The CPLEAR Collaboration, Phys. Lett. B369 (1996) 367 14. The CPLEAR Collaboration, Phys. Lett. B444 (1998) 43 15. The CPLEAR Collaboration, Eur. Phys. J. C5 (1998) 389 16. P. Kokkas, talk presented at the XXIX International Conference on High Energy Physics, Vancouver, July, 1998 17. Particle Data Group, Eur. Phys. J. C3 (1998) 1 18. M. Ciuchini, Nucl. Phys. Proc. Suppl. 59 (1997) 149, and references therein 19. S. Bosch, A.J. Buras, M. Gorbahn, S. J¨ager, M.Jamin, M.E. Lautenbacher and L. Silvestrini, Standard model confronting new results for ε′ /ε, hep-th/9904408 20. S. Bertolini, M.Fabbrichesi and J.O. Eeg, Estimating ε′ /ε, hep-th/9802405 21. The KTEV Collaboration, http://fnphyx-www.fnal.gov/experiments/ktev/epsprime/epsprime.html 22. Y.-Y. Keum, U. Nierste and A.I. Sanda, A short look at ε′ /ε, hep-th/9903230 23. X.-G. He, Contribution to ε′ /ε from anomalous gauge couplings, hep-th/9903242 24. A. Masiero and H. Murayama, Can ε′ /ε be supersymmetric?, hep-th/9903363 25. F. Benatti and R. Floreanini, Phys. Lett. B428 (1998) 149 26. G. D’Ambrosio, G. Isidori and A. Pugliese, CP and CP T measurements at DaΦne, in The Second Da Φne Physics Handbook, L. Maiani, G. Pancheri and N. Paver, eds., (INFN, Frascati, 1995) 11